# nLab hyperbolic manifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

A hyperbolic space is the analog of a Euclidean space as one passes from Euclidean geometry to hyperbolic geometry. The generalization of the concept of hyperbolic plane to higher dimension.

A hyperbolic manifold is a geodesically complete Riemannian manifold $(X,g)$ of constant sectional curvature $-1$.

Of particular interest are hyperbolic 3-manifolds.

## Properties

### Conformal flatness

Every hyperbolic manifold is a conformally flat manifold.

(e.g. Long-Reid 00, p. 4)

### Zeta functions

There are canonical zeta functions associated with suitable (odd-dimensional) hyperbolic manifolds, see at Selberg zeta function and Ruelle zeta function.

### Mostow rigidity theorem

The Mostow rigidity theorem states that every hyperbolic manifold of dimension $\geq 3$ and of finite volume is uniquely determined by its fundamental group.

## Examples

A Riemannian manifold

• with zero sectional curvature is a Euclidean manifold?;

• with +1 sectional curvature is an elliptic manifold?